The low-frequency spectrum of small Helmholtz resonators
We analyse the spectrum of the Laplace operator in a complex geometry, representing a small Helmholtz resonator. The domain is obtained from a bounded set Ω⊂Rn by removing a small obstacle Σ ⊂Ω of size >0. The set Σ essentially separates an interior domain Ωεinn (the resonator volume) from an ext...
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Veröffentlicht in: | Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2015-02, Vol.471 (2174), p.20140339 |
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description | We analyse the spectrum of the Laplace operator in a complex geometry, representing a small Helmholtz resonator. The domain is obtained from a bounded set Ω⊂Rn by removing a small obstacle Σ ⊂Ω of size >0. The set Σ essentially separates an interior domain Ωεinn (the resonator volume) from an exterior domain Ωεout, but the two domains are connected by a thin channel. For an appropriate choice of the geometry, we identify the spectrum of the Laplace operator: it coincides with the spectrum of the Laplace operator on Ω, but contains an additional eigenvalue με−1. We prove that this eigenvalue has the behaviour μ V L /A , where V is the volume of the resonator, L is the length of the channel and A is the area of the cross section of the channel. This justifies the well-known frequency formula ωHR=c0A/(LV) for Helmholtz resonators, where c0 is the speed of sound. |
doi_str_mv | 10.1098/rspa.2014.0339 |
format | Article |
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This justifies the well-known frequency formula ωHR=c0A/(LV) for Helmholtz resonators, where c0 is the speed of sound.</description><identifier>ISSN: 1364-5021</identifier><identifier>EISSN: 1471-2946</identifier><identifier>DOI: 10.1098/rspa.2014.0339</identifier><language>eng</language><publisher>The Royal Society Publishing</publisher><subject>Complex Geometry ; Helmholtz Resonator ; Sound Attenuator ; Spectral Properties Of The Laplace Operator</subject><ispartof>Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences, 2015-02, Vol.471 (2174), p.20140339</ispartof><rights>2014 The Author(s) Published by the Royal Society. 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The domain is obtained from a bounded set Ω⊂Rn by removing a small obstacle Σ ⊂Ω of size >0. The set Σ essentially separates an interior domain Ωεinn (the resonator volume) from an exterior domain Ωεout, but the two domains are connected by a thin channel. For an appropriate choice of the geometry, we identify the spectrum of the Laplace operator: it coincides with the spectrum of the Laplace operator on Ω, but contains an additional eigenvalue με−1. We prove that this eigenvalue has the behaviour μ V L /A , where V is the volume of the resonator, L is the length of the channel and A is the area of the cross section of the channel. This justifies the well-known frequency formula ωHR=c0A/(LV) for Helmholtz resonators, where c0 is the speed of sound.</description><subject>Complex Geometry</subject><subject>Helmholtz Resonator</subject><subject>Sound Attenuator</subject><subject>Spectral Properties Of The Laplace Operator</subject><issn>1364-5021</issn><issn>1471-2946</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><recordid>eNp1jzFPwzAQhS0EEqWwMucPJNzFcRyPqAKKVImlzJbtXtRWSRzsRCj8ehKVleneDd_T-xh7RMgQVPUUYm-yHLDIgHN1xVZYSExzVZTXc-ZlkQrI8ZbdxXgGACUquWLV_khJ47_TOtDXSJ2bktiTG8LYJr5OYmuaJtlS0x59M_wkgaLvzOBDvGc3tWkiPfzdNft8fdlvtunu4-1987xLHc_5kDpnXFnYCgyJGgEOXFgkQ1aBkHO0VpSlldZifkBZc1QAlmYLjkIqQXzNskuvCz7GQLXuw6k1YdIIevHWi7devPXiPQP8AgQ_zcO8O9Ew6bMfQze__1G_NQNd0Q</recordid><startdate>20150208</startdate><enddate>20150208</enddate><creator>Schweizer, B.</creator><general>The Royal Society Publishing</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20150208</creationdate><title>The low-frequency spectrum of small Helmholtz resonators</title><author>Schweizer, B.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c323t-ccac64b80ae5f100d35b1eaeb90575b1bb566b7bb12d17f31900be201315795e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Complex Geometry</topic><topic>Helmholtz Resonator</topic><topic>Sound Attenuator</topic><topic>Spectral Properties Of The Laplace Operator</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Schweizer, B.</creatorcontrib><collection>CrossRef</collection><jtitle>Proceedings of the Royal Society. 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The set Σ essentially separates an interior domain Ωεinn (the resonator volume) from an exterior domain Ωεout, but the two domains are connected by a thin channel. For an appropriate choice of the geometry, we identify the spectrum of the Laplace operator: it coincides with the spectrum of the Laplace operator on Ω, but contains an additional eigenvalue με−1. We prove that this eigenvalue has the behaviour μ V L /A , where V is the volume of the resonator, L is the length of the channel and A is the area of the cross section of the channel. This justifies the well-known frequency formula ωHR=c0A/(LV) for Helmholtz resonators, where c0 is the speed of sound.</abstract><pub>The Royal Society Publishing</pub><doi>10.1098/rspa.2014.0339</doi><oa>free_for_read</oa></addata></record> |
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subjects | Complex Geometry Helmholtz Resonator Sound Attenuator Spectral Properties Of The Laplace Operator |
title | The low-frequency spectrum of small Helmholtz resonators |
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